Solve by Factoring x^6+x^3-6=0

$x^{6} + x^{3} - 6 = 0$

Step 1

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

${(x^{3})}^{2} + x^{3} - 6 = 0$

Step 2

Let $u = x^{3}$ . Substitute $u$ for all occurrences of $x^{3}$ .

$u^{2} + u - 6 = 0$

Step 3

Factor $u^{2} + u - 6$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $1$ .

$- 2, 3$

Write the factored form using these integers.

$(u - 2) (u + 3) = 0$

Step 4

Replace all occurrences of $u$ with $x^{3}$ .

$(x^{3} - 2) (x^{3} + 3) = 0$

Step 5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{3} - 2 = 0$

$x^{3} + 3 = 0$

Step 6

Set $x^{3} - 2$ equal to $0$ and solve for $x$ .

Set $x^{3} - 2$ equal to $0$ .

$x^{3} - 2 = 0$

Solve $x^{3} - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

$x^{3} = 2$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{2}$

Step 7

Set $x^{3} + 3$ equal to $0$ and solve for $x$ .

Set $x^{3} + 3$ equal to $0$ .

$x^{3} + 3 = 0$

Solve $x^{3} + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$x^{3} = - 3$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{- 3}$

Simplify $\sqrt[3]{- 3}$ .

Rewrite $- 3$ as ${(- 1)}^{3} \cdot 3$ .

Rewrite $- 3$ as $- 1 (3)$ .

$x = \sqrt[3]{- 1 (3)}$

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{(- 1)}^{3} \cdot 3}$

Pull terms out from under the radical.

$x = - 1 \sqrt[3]{3}$

Rewrite $- 1 \sqrt[3]{3}$ as $- \sqrt[3]{3}$ .

$x = - \sqrt[3]{3}$

Step 8

The final solution is all the values that make $(x^{3} - 2) (x^{3} + 3) = 0$ true.

$x = \sqrt[3]{2}, - \sqrt[3]{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt[3]{2}, - \sqrt[3]{3}$

Decimal Form:

$x = 1.25992104 \dots, - 1.44224957 \dots$

Solve by Factoring x^6+x^3-6=0

Related Questions

Support LCMGCF.com by disabling your adblocker.